Effective Bogomolov conjecture for non-isotrivial fibrations

Let $X$ be a smooth projective surface over a field $k$, let $Y$ be a smooth projective curve over $k$, and let $f:X\to Y$ be a semistable fibration whose smooth generic fiber $C/K$ has genus $\bar{g}\geq 2$. Let $K$ have algebraic closure $\overline{K}$, and let $r_C(P)$ be defined from the Néron–Tate seminorm on $\operatorname{Pic}^0(C)(\overline{K})$. Effective Bogomolov conjecture. If $f$ is non-isotrivial, then there exists an effectively calculable positive number $r_0$ such that

$$\text{inf}_{P\in\operatorname{Pic}^0(C)(\overline{K})}r_C(P)\geq r_0.$$

The claim strengthens Bogomolov's conjecture by requiring a uniform and effectively computable positive lower bound; the source gives no resolution status.